Analyzing F'(x) = (x-5)^2(x+7) Critical Points, Intervals, And Extrema

by ADMIN 71 views

In calculus, derivatives serve as a powerful tool for understanding the behavior of functions. By analyzing the first derivative, we can determine critical points, intervals of increasing and decreasing behavior, and potential local extrema. This article delves into a comprehensive analysis of a function whose derivative is given by f'(x) = (x-5)^2(x+7), addressing key questions about its critical points, intervals of increase and decrease, and the existence of local extrema.

a. Identifying Critical Points of f

Critical points, the cornerstone of function analysis, are the x-values where the derivative of a function is either zero or undefined. These points hold immense significance as they pinpoint potential locations of local maxima, local minima, or points of inflection within the function's graph. To embark on the quest for the critical points of our function f, we must first set its derivative, f'(x), equal to zero and embark on the algebraic journey to solve for x. In simpler terms, we're seeking the specific x-values that render the tangent line to the function's curve horizontal, a telltale sign of a critical point lurking nearby.

Given the derivative f'(x) = (x-5)^2(x+7), we set it to zero:

(x-5)^2(x+7) = 0

This equation reveals two factors that can potentially lead to a zero product: (x-5)^2 and (x+7). Let's dissect each factor individually:

  1. (x-5)^2 = 0

    Taking the square root of both sides, we get:

    x-5 = 0

    Adding 5 to both sides, we arrive at:

    x = 5

  2. (x+7) = 0

    Subtracting 7 from both sides, we obtain:

    x = -7

Therefore, the critical points of the function f are x = 5 and x = -7. These critical points act as signposts, guiding us to potential turning points in the function's trajectory, where it might transition from increasing to decreasing or vice versa. They are the key to unlocking a deeper understanding of the function's local behavior, revealing the presence of peaks, valleys, or inflection points that shape its overall form. In the subsequent sections, we will leverage these critical points to map out the intervals where the function ascends or descends and to pinpoint the precise locations of any local extrema that may grace its path.

b. Determining Intervals of Increase and Decrease

To determine where the function f is increasing or decreasing, we analyze the sign of the derivative f'(x) in the intervals defined by the critical points. The critical points, x = -7 and x = 5, divide the real number line into three intervals: (-∞, -7), (-7, 5), and (5, ∞). By examining the sign of f'(x) within each interval, we can deduce whether the function is climbing uphill (increasing) or sliding downhill (decreasing). Think of it as a compass guiding us through the function's terrain, revealing its ascents and descents.

We will test a value within each interval to determine the sign of f'(x):

  1. Interval (-∞, -7):

    Let's choose x = -8.

    f'(-8) = (-8-5)^2(-8+7) = (169)(-1) = -169

    Since f'(-8) < 0, the function f is decreasing on the interval (-∞, -7). Imagine the function as a skier descending a slope, its path steadily declining as it traverses this interval.

  2. Interval (-7, 5):

    Let's choose x = 0.

    f'(0) = (0-5)^2(0+7) = (25)(7) = 175

    Since f'(0) > 0, the function f is increasing on the interval (-7, 5). Now picture the skier ascending a hill, exerting effort as they climb towards a higher elevation. The function mirrors this effort, its values steadily rising within this interval.

  3. Interval (5, ∞):

    Let's choose x = 6.

    f'(6) = (6-5)^2(6+7) = (1)(13) = 13

    Since f'(6) > 0, the function f is increasing on the interval (5, ∞). The skier continues their upward journey, the function maintaining its ascent as it ventures further along the x-axis.

In summary, the function f is decreasing on the interval (-∞, -7) and increasing on the intervals (-7, 5) and (5, ∞). This knowledge paints a vivid picture of the function's dynamic behavior, revealing its periods of descent and ascent. By identifying these intervals, we gain a deeper understanding of how the function undulates and evolves across its domain, paving the way for a more complete analysis of its properties.

c. Identifying Local Extrema

Local extrema, the peaks and valleys in a function's landscape, hold immense significance in understanding its behavior. These points, where the function reaches a local maximum or minimum value, mark critical transitions in its trajectory. The first derivative test serves as our compass in this exploration, guiding us to identify these pivotal points by analyzing the sign changes in the derivative around the critical points. It's like deciphering a map to locate the highest peaks and deepest valleys in a terrain.

We examine the sign changes of f'(x) around the critical points x = -7 and x = 5:

  1. At x = -7:

    The derivative f'(x) changes from negative to positive as x passes through -7. This indicates that the function f changes from decreasing to increasing at x = -7. Therefore, there is a local minimum at x = -7. Imagine a valley in the function's graph, where it dips to its lowest point before ascending again. This local minimum marks the bottom of that valley.

  2. At x = 5:

    The derivative f'(x) does not change sign as x passes through 5. It remains positive on both sides of x = 5. This means the function f continues to increase through x = 5 without changing direction. Therefore, there is no local extremum at x = 5. Picture a gentle slope in the function's graph, where it continues its ascent without forming a peak or valley. The absence of a sign change in the derivative signals the absence of a local extremum at this point.

In conclusion, there is a local minimum at x = -7, and there is no local maximum or minimum at x = 5. This information enhances our understanding of the function's local behavior, pinpointing the precise locations where it attains its local minimum value. By identifying these extrema, we gain a more nuanced appreciation of the function's shape and characteristics, completing our exploration of its critical points, intervals of increase and decrease, and the presence of local peaks and valleys.

Final Thoughts

By analyzing the derivative f'(x) = (x-5)^2(x+7), we've successfully identified the critical points of the function f, determined the intervals where it increases and decreases, and located its local extrema. This comprehensive analysis demonstrates the power of derivatives in unraveling the behavior of functions. The critical points act as guideposts, leading us to the function's potential turning points, while the sign of the derivative reveals its ascents and descents. By piecing together these insights, we gain a deeper understanding of the function's overall shape and characteristics, paving the way for further exploration and analysis.

This exploration provides a robust framework for analyzing functions using derivatives, applicable to a wide range of mathematical problems and real-world applications. By mastering these techniques, one can confidently navigate the complexities of functions and unlock their hidden patterns and behaviors.